{smcl}
{* 10dec2021}{...}
{cmd:help mata mm_hl()}
{hline}

{title:Title}

{p 4 17 2}
{bf:mm_hl() -- Robust location, scale, and skewness estimators based on pairwise combinations}

{title:Syntax}

{pstd}
Hodges-Lehmann location estimator

{p 8 24 2}
{it:real scalar}{bind: }
{cmd:mm_hl(}{it:X} [{cmd:,} {it:w}{cmd:,} {it:fw}{cmd:,} {it:naive}]{cmd:)}

{pstd}
Qn scale coefficient

{p 8 24 2}
{it:real scalar}{bind: }
{cmd:mm_qn(}{it:X} [{cmd:,} {it:w}{cmd:,} {it:fw}{cmd:,} {it:naive}]{cmd:)}

{pstd}
Medcouple skewness measure

{p 8 24 2}
{it:real scalar}{bind: }
{cmd:mm_mc(}{it:X} [{cmd:,} {it:w}{cmd:,} {it:fw}{cmd:,} {it:naive}]{cmd:)}

{p 4 8 2}
where

{p 13 18 2}{it:X}:  {it:real colvector} containing data
{p_end}
{p 13 18 2}{it:w}:  {it:real colvector} containing weights
{p_end}
{p 12 18 2}{it:fw}:  {it:real scalar} indicating that weights are frequency weights
{p_end}
{p 9 18 2}{it:naive}:  {it:real scalar} requesting the naive estimator (slow)


{title:Description}

{pstd}
{cmd:mm_hl()}, {cmd:mm_qn()}, and {cmd:mm_mc()} compute robust estimates of
location, scale, and skewness based on pairwise combinations of
observations, as suggested by Hodges and Lehmann (1963), Rousseeuw and Croux (1993),
and Brys et al. (2004), respectively. Note that {cmd:mm_mc()} can also be used to
compute robust tail-weight (kurtosis) measures suggested by
Brys et al. (2006) (see examples below).

{pstd}
Argument {it:X} specifies the data for which the measures be computed.

{pstd}
Argument {it:w} specifies weights associated with the observations in {it:X}.
Omit {it:w} or specify {it:w} as 1 to obtain unweighted results.

{pstd}
Argument {it:fw}!=0 indicates that the weights are to be interpreted as
frequency weights. The default is to interpret the weights as sampling weights.

{pstd}
Argument {it:naive}!=0 requests that an algorithm enumerating all pairs be
used to compute the measures. The default is to use variants of the fast
algorithm proposed by Johnson and Mizoguchi (1978) (also see Croux and
Rousseeuw 1992, Brys et al. 2004). The only purpose of the {it:naive} argument
is to provide a way to confirm that the results from the fast algorithm are
correct. The naive algorithm will only be feasible in small datasets.


{title:Examples}

{pstd}
Generate standard normally distributed data:

        {com}: x = rnormal(1000, 1, 0, 1){txt}

{pstd}
Classical mean, standard deviation, and skewness:

        {com}: mean(x), sqrt(variance(x)), mean((x:-mean(x)):^3)/mm_variance0(x)^(3/2)
        {res}       {txt}           1              2              3
            {c TLC}{hline 46}{c TRC}
          1 {c |}  {res} .0042933326    1.031374246   -.0582095664{txt}  {c |}
            {c BLC}{hline 46}{c BRC}{txt}

{pstd}
Robust location, scale, and skewness:

        {com}: mm_hl(x), mm_qn(x), mm_mc(x)
        {res}       {txt}          1             2             3
            {c TLC}{hline 43}{c TRC}
          1 {c |}  {res}.0102320223   1.041665907   .0321525454{txt}  {c |}
            {c BLC}{hline 43}{c BRC}{txt}

{pstd}
Add 5% contamination at x = 10:

        {com}: x[|1\50|] = J(50, 1, 10){txt}

{pstd}
Classical mean, standard deviation, and skewness:

        {com}: mean(x), sqrt(variance(x)), mean((x:-mean(x)):^3)/mm_variance0(x)^(3/2)
        {res}       {txt}          1             2             3
            {c TLC}{hline 43}{c TRC}
          1 {c |}  {res}.5062068281   2.400310653   2.975298653{txt}  {c |}
            {c BLC}{hline 43}{c BRC}{txt}

{pstd}
Robust location, scale, and skewness:

        {com}: mm_hl(x), mm_qn(x), mm_mc(x)
        {res}       {txt}          1             2             3
            {c TLC}{hline 43}{c TRC}
          1 {c |}  {res}.1127161045    1.15192769   .0534097435{txt}  {c |}
            {c BLC}{hline 43}{c BRC}{txt}

{pstd}
Compute left and right medcouple tail weight measures as suggested by Brys et al. (2006):

        {com}: med = mm_median(x)
        {res}
        {com}: lmc = -mm_mc(select(x, x:<med))
        {res}
        {com}: rmc =  mm_mc(select(x, x:>med))
        {res}
        {com}: lmc, rmc
        {res}       {txt}          1             2
            {c TLC}{hline 29}{c TRC}
          1 {c |}  {res} .143736566   .2359231771{txt}  {c |}
            {c BLC}{hline 29}{c BRC}{txt}


{title:Conformability}

    All functions:
             {it:X}:  {it:n x} 1
             {it:w}:  {it:n x} 1 or 1 {it:x} 1
            {it:fw}:  1 {it:x} 1
         {it:naive}:  1 {it:x} 1
        {it:result}:  1 {it:x} 1


{title:Diagnostics}

{pstd}
The functions return {cmd:.} (missing) if {it:X} is void.

{pstd}
The functions return error if {it:X} or {it:w} contain missing.

{pstd}
The functions return error if {it:w} contains negative values.

{pstd}
The functions return error {it:fw}!=0 is specified and {it:w} is non-integer.


{title:Source code}

{pstd}
{help moremata11_source##mm_hl:mm_hl.mata}


{title:References}

{phang}
    Brys, G., M. Hubert, A. Struyf (2004). A Robust Measure of Skewness.
    Journal of Computational and Graphical Statistics 13(4): 996-1017.
    {p_end}
{phang}
    Brys, G., M. Hubert, A. Struyf (2006). Robust measures of tail weight.
    Computational Statistics & Data Analysis 50: 733-759.
    {p_end}
{phang}
    Croux, C., P. J. Rousseeuw (1992). Time-efficient algorithms
    for two highly robust estimators of scale. P. 411-428 in: Y. Dodge and J.
    Whittaker (eds.). Computational Statistics. Heidelberg: Physica-Verlag.
    {p_end}
{phang}
    Johnson, D. B., T. Mizoguchi (1978). Selecting the {it:K}th element
    in {it:X} + {it:Y} and {it:X}_1 + {it:X}_2 + ... + {it:X}_{it:m}. SIAM
    Journal on Scientific Computing 7(2): 147–153.
    {p_end}
{phang}
    Hodges, Jr., J. L., E. L. Lehmann (1963). Estimates of location based on
    rank tests. Annals of Mathematical Statistics 34(2): 598-611.
    {p_end}
{phang}
    Rousseeuw, P. J., C. Croux (1993). Alternatives to the Median
    Absolute Deviation. Journal of the American Statistical Association
    88(424): 1273-1283.
    {p_end}


{title:Author}

{pstd}
Ben Jann, University of Bern, ben.jann@soz.unibe.ch


{title:Also see}

{p 4 13 2}
Online:  help for
{helpb moremata}
